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Question #43120

Difficulty: 8/10 | 🟢 Free

The Question

An isotropic point source of light emitting a total time-averaged electromagnetic power $P$ (in $\text{W}$ ) is placed at a depth $h$ (in $\text{m}$ ) below the flat surface of a transparent, non-absorbing liquid of refractive index $n$ . A small planar detector is placed in the air (refractive index $n_{air} = 1$ ) at a height $H$ (in $\text{m}$ ) directly above the source, with its surface parallel to the liquid surface. Assuming paraxial rays and neglecting any reflection losses at the liquid-air interface, what is the magnitude of the time-averaged Poynting vector $\langle S \rangle$ measured at the detector?

Options

A.
$\frac{P}{4\pi (h + nH)^2}$
B.
$\frac{n^2 P}{4\pi (h + nH)^2}$
C.
$\frac{P}{4\pi n (h + nH)^2}$
D.
$\frac{P}{4\pi (h + H)^2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Apparent Depth/Shift|Poynting Vector and Intensity|Binomial Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the conservation of energy flux and the correct application of the Poynting vector when light crosses a dielectric interface.

Students commonly fall into the trap of treating the virtual image (located at the apparent depth $h/n$ ) as an isotropic point source with the same total power $P$ .

This leads them to incorrectly calculate the intensity as $I = P / [4\pi(H + h/n)^2]$ , which simplifies to Option B.

They fail to account for the fact that refraction alters the solid angle of the beam ( $d\Omega_{air} = n^2 d\Omega_{liquid}$ ), meaning the power per unit solid angle decreases by a factor of $n^2$ .

The correct approach requires tracking the actual cross-sectional area of the diverging beam to find the true energy density and intensity.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Apparent Depth/Shift
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free